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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeMay 24th 2010

    I created Cantor space to record its definition as a locale, but goodness knows there is no end to what might be written about it.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 24th 2010

    I cross-linked Cantor space from (newly created) Examples-sections at topological space and locale and topology - contents

    Eventually hopefully this sidebar for topology is expanded to something that reflects the scope of the relevant nLab articles

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 21st 2015

    I added a bit more to Cantor space, including the abstract characterization up to homeomorphism (which was oddly missing, since what was there seemed to be leading right up to that point). While I was at it, I created perfect space (with perfect set redirected to it).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2017

    I gave Cantor space more of an Idea-section. Then I expanded the discussion at As a subspace of the real line with full detail. The same discussion I also copied over to Tychonoff product in this example.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 8th 2017

    I’m interested in the last sentence of the Idea section. In what way is Cantor space used to construct the Peano curve?

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2017

    Worth adding the coalgebraic description? What is it, the terminal coalgebra for the endofunctor on TopTop, XX+XX \mapsto X + X?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2017
    • (edited May 8th 2017)

    Todd, sorry, I should have been more explicit. Maybe I should write “may be used to neatly organize the construction” rather than “may be used to construct”: I am thinking of picking a continuous surjection CantortCantor×CantorCantor \overset{t}{\to} Cantor \times Cantor (e.g. unshuffle), then observing that there is easily a continuous surjection s:Cantor[0,1]s \colon Cantor \to [0,1] and, with a tad more work, that every continuous function from CantorCantor to a linear space may be extended along the defining embedding Cantor[0,1]Cantor \hookrightarrow [0,1] (by linear interpolation). Then applying this extension to the surjection CantorCantor×tCantor(s,s)[0,1]×[0,1]Cantor \to Cantor \overset{t}{\times} Cantor \overset{(s,s)}{\to} [0,1] \times [0,1] gives the desired continuous surjection.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 8th 2017

    Well, I’ll be. That’s rather nice, Urs. Never saw that before (and see nothing wrong with it).

    David: that’s right.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2017

    Ok, I’ll put it in.

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 9th 2017

    Being much taken with the simplicity of this Peano curve as sketched by Urs #7, I looked around and saw this is called the Lebesgue space-filling curve, which has another nice properties such as being differentiable almost everywhere. It’s obviously similar in flavor to the Cantor-Lebesgue function.

    Anyway, I went ahead and bashed out the construction at Peano curve, with a proof of continuity. It was just a quick and dirty job, which I may see about cleaning up later. Not many cross-links were inserted. (It’s now bedtime for me.)

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2017

    Thanks, Todd! That’s very nice. I didn’t know that Lebesgue’s name is associated with this.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 9th 2017

    I did some more jiggering with Peano curve, which then led me to add to Cantor space a proof of the Hausdorff-Alexandroff theorem, which says that every compact metric space is a continuous image of Cantor space.

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